Mechanics of Fluids and Pressure - Part IV

3. Pascalís law of transmission of pressure in fluids
Pascalís law is a useful application and extension of the phenomena of liquids seeking their levels automatically. Take a vessel that has many limbs and an airtight piston at the centre.

Fill the vessel with liquid. Press the piston, you will notice that the liquid level has risen in the other limbs equally. This implies that the pressure applied on the piston is transmitted immediately in other parts of the liquid. Remember that an increase in height means increase in pressure. (It is important to note that the liquid has to be a non-compressible fluid. The same will not be applicable to gases.)

Pascalís Law states that pressure exerted at a point in an enclosed (and non-compressible) liquid is transmitted equally in all directions. The pressure is transmitted undiminished.

Applications of Pascalís law are plenty. The most common use this has is in hydraulic lifts, car brakes etc. How this is achieved is shown in the diagram below.

Two cylinders are connected in an airtight fashion. The arrangement is filled with liquid. The piston 1 in one cylinder is smaller than piston 2 in  other cylinder.  On the smaller piston, a load (force) of W₁ is applied.

                                                                             W₁
The pressure on the liquid in piston 1 is P₁      =                  A₁ = area of piston1
                                                                             A₁

According to the Pascalís Law,  the same pressure is transmitted undiminished to piston 2. Thus at piston 2 the pressure P₂.

                                      ?
P₁    =  P₂            =                            ? = force experience at piston 2
                                     A₂

Since A₂   > A₁, the load (force) experienced at piston 2  will be larger than W₁. This force will be in the opposite direction to the force applied on piston 1.

If     A₁/A₂  = 10, then a force which is 10 times larger than the force applied on piston 1 is experienced at piston 2.

Now you can appreciate how car breaks function. The break system is filled with special break fluid. A small pressure break paddle is able to stop a moving car which has such a large mass than the break paddle itself!!

4. Atmospheric pressure and its measurement
The air or the atmosphere around the earth extends upwards up to about 300 km from the sea level. The density of air is maximum at the sea level and becomes rarer as we go higher up (due to gravitational pull of the earth, air is more dense close to the surface of the earth). The air exerts pressure on surfaces that it is in contact with. At sea level, the air pressure is 101325 N/m².  Atmospheric pressure decreases as we go to higher and higher altitudes, since the density of air also decreases. Atmospheric pressure at a given point is same in all directions. This is similar to the case of liquids. The entire weather system of the earth depends on the movement of air. Air moves from high-pressure region to low pressure region, in order to equalize pressure.

We can demonstrate two simple experiments in order to understand the presence of atmospheric pressure.

Experiment 1 :  Take a tin can and heat it with its cork in the open position. The heat will make the air inside hot. The hot air will leave the container. Now quickly close the container. You will notice that the tin can collapses. Why does this happen? The air inside the tin container is expelled partially on heating. When you close the container, there is no way for the air outside to enter the tin. The air pressure inside is less than the air pressure outside the tin. The air outside the tin presses against its surface. This crushes the tin.  The surface area of the tin is reduced, till such time that the pressure inside becomes same as the pressure outside.

The experiment shows that air exerts pressure.

Experiment 2 :  Take a glass and fill it with water up to the brim. No air gap should remain between the water level and the rim of the glass. Cover the glass with a postcard or a cardboard piece. Invert the glass quickly. You will notice that the cardboard and water are held in place. This implies that some force is acting on the cardboard from below, which is able to hold the weight of water on it. The force is the atmospheric pressure acting in the upward direction.

The experiment also shows that air exerts pressure.

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